DESIGN AND CONSTRUCTION OF HYDRAULIC FLUME AND BACKWATER EFFECTS OF SEMI- CIRCULAR CONSTRICTIONS IN A SMOOTH RECTANGULAR CHANNEL Progress Report- Ho* 2 ID; K« B* V.'oodSj, Director Joint Highway Research Project FRGMs K» lit Mchael s Assistant Director Joint Highway iteseareh Project January 21 9 I960 Ellas 9-S-2 Project: C«-36-»62I Attached is a progress report entitle^ "Design and Construction of Kydraulic Fluss and Backwater Bffeots of SeHii«*Cireular Constrictions in a SiEOoth Rectangular Channel"* This report has hs&n prepared by Mr« He J» 0wsn 5 grsdu&ta research assistant oa cm 1 staff 9 un<fer the direction cf J* W, Delleur* Mr* Osjan also utilised the report as his thesis in psrtial fuifiilnsnt for the r®o;aireiisnt of the M»8»C*E* degree* The rsaterial reported in this report is a suuseary of the programs ■that has occurred on the Hydraulics of Arch Bridges Project ishich is being conducted in cooperation t&th the Indiana State Highway Bepajpfessnt and the U» S« Bureau, of Public Roads* Copies of this report *411 a2s© be distributed to the State Highway Department and the Bareaa of Public Roads for thsir review and coKsneats* t The report is presented to the Board for the record* Respectfully subaittedg, HZcisksse Attaehnsnt cc: F. Ashbaechep J e R* Cooper W» L« Bolsh ¥. H« Gcets P. P. Ba-gsy G. A. Ea&l&ns (M» B, Scott) G» A. Leonards H» L. Michael, Secretary J. F* 2feX&ughli» R. B» Miles R> 3, Kills C» S* Vogelgosang J. L. Waling J* E. Wilson F.» J. Yodsy Progress Report Ko» 2 Design and Construction of Hydraulic rTcaae and Backwater Effects of Sead-Circular Constrictions in A Smooth Eectangular Chamol Graduate Assistant Joint Highly Research Project Project Bo, C~36«42B nis No« 9«&*a Pardw University Lafayette, Indians .January 21, I960 A J I'd I U i , hti. IG-M.CJN ! 3 The author wishes to acknowledge and thank Dr. J. W. Delleur whose guidence and help was invaluable. Appreciation must also be expressed by the author to his wife .Pat. With- out her help, the preparation of the manuscript would have been difficult. Digitized by the Internet Archive in 2011 with funding from LYRASIS members and Sloan Foundation; Indiana Department of Transportation http://www.archive.org/details/designconstructiOOowen TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v LIST OF TABLES vii ABSTRACT viii INTRODUCTION 1 REVIEW OF THE LITERATURE 2 DESIGN AND CONSTRUCTION OF THE APPARATUS 5 Testing Flume 5 Preliminary Computations 5 Length of Backwater Profile 7 Design Procedure 10 Dead Load 12 Live Load 12 Tail Gate 14 Forebay 14 Slope Control Mechanism 15 Construction 17 PREPARATION FOR TESTING 18 Slope Calibration 18 Water Supply and Measurement System 19 Venturi Calibration 21 MODEL TESTING 22 Model Construction 22 Free Surface Measurement 23 Uniform Flow Calibration 24 Testing 25 Test Results 27 ARY OF RESULTS AND CONCLUSIONS 32 Flume Design 32 Testing 32 IV BIBLIOGRAPHY^ ' . .' 34 APPENDIX A NOTATIONS 36 APPENDIX B EQUATIONS OP PLOW 39 APPJSMDIX J jj'IGUREo AND TABLES 41 LIST OF ILLUSTRATIONS Figure Title Page 1. Idealized Stream cross section 41 2. Conveyance versus Depth Curve 42 3. Length of Regain 43 4. Flume Construction 44 5. Flume and Models 45 6. Tail gate Construction 45 7. Plan View of Jacks and Gearing 46 8. Jack Detail 47 9. Slope Jalibration Curve For Hydraulic Testing Flume 48 10. Apparatus Arrangement 49 11. Calibration Curve for 3 Inch Venturi 50 12. Calibration Curve for 6 Inch Venturi 51 13. Calibration Curve for 6 Inch Venturi 52 14. Flow in Rectangular Channels with Semi- circular Constrictions-Comparison of two and Three Dimensional Cases 53 15. Model Construction 5^ 16. Instrument Carriage 55 17. Model in Place 55 18. Model in Place 56 19. Model in Place 56 20. Normal Depth Versus Slope for Testing Flume ... 57 21. Slope Versus Gate iieigiith for Testing Flume ... 57 VI 22. Superelevation Versus Kineticity . . . * 53 23. Discharge Coefficient Versus Kineticity 59 24. Friction Factor Versus Reynolds iiumber 60 25. Friction Factor Versus Reynolds Number 61 26. Backwater Ratio Versus Contraction Ratio 62 27. Definition Sketch 53 Vll LIST OP TABLES Table Page 1. Observed and Calculated Data 64 Vlll ABSTRACT a. James Owen, lw.S.C.E. Purdue University, January I960. "Design and construction of Hydraulic Flume and .backwater Effects of semi-Circular Jonstrictions in a smooth Rectangular Ohannel". Major Professor: Dr. J. W. Delleur. The purpose of this study was to investigate the hydraulics of semi-circular constrictions in smooth rect- angular channels. To this end a large part of this work consisted of the design and construction of a hydraulic flume 54 feet long and 5 feet wide. This study is part of a general program sponsored by the State Highway De- partment of Indiana and the U.S. Bureau of Public Roads at Purdue university on the hydraulics of River Flow Under Arch bridges. INTRODUCTION This project was initiated in the Hydraulics laboratory of Purdue University by the Indiana State Highway Departi i in cooperation with the U.S. Bureau of Public Roads, to study the backwater and regain phenomena associated with arched constrictions such as sre presented by arch bridges. The problem was an- roached by both theoretical analysis and model study. The preliminary analysis of the general problem was made by Mr. S. T. Husain^ ' . Mr. A. A. Sooky^ ' derived exact and approximate equations for the two dimensional sharp edged constrictions, and carried out a preliminary testing program in a small flume. The present study reports on the design and construction of the flume for the main testing program. The design was begun in July, 1958 and construction was started in March, 1959. The flume and its appurtenances were constructed and operative in August, 1959. Also in- cluded is the testing of two dimensional semi-circular con- strictions carried out from September 1959 to November 1959. Superscripts refer to bibliography at end of text. ttdvlliiW Ui)' THE LITS^ATUftE i.ane v , in 1915 and 1916, performed several exper- iments on contracted openings of various tyres. The full series of tests was not completed but data was collected and evaluated on the following: 1. Sharp edged vertical contractions 2. A rounded edge vertical contraction 3. A short flume with rounded entrance 4-. a short flume with sharp corner entrance 5. An expanding flume The paper describes the apparatus and proceiure util- ized as well as the results obtained. Lane found that the JVeisbach formula { Q = Cp b )/2j~ (% H '* + i/ H% ] where H is the total drop of the water in passing through the contraction including the change in the velocity head and y is the average downstream depth. ) wis most nearly in agreement with the experimental results obtained in the case of the sharp edged contractions. Kindsvater aril darter^ ' attempted to correlate labor- atory results with field data obtained by the USGS. A gen- eral equation was developed for discharge through a con- tracted opening: Q.-Qphy ^E^h^T^fj where y * See list of symbols in Appendix A is the downstream depth. This expression contained a dis- charge coefficient which was determined for several cases. The base curves of the discharge coefficient in terms of the contraction ratios and length of the constriction were presented for a standard condition of square abutment and Froude number of 0.5. The base curves are supplemented by auxiliary curves modifying the coefficient due to: 1. Frou.de number different from standard 2. Upstream corner rounding 3. Eccentricity 4. Skew 5. Channel contraction 6. Chamfers of the abutments 7. oide depths 8. Abutment side slopes 9. Submergence 10. Bridge piers and piles Tracy and barter ' made a laboratory study in 195^ of the flow through contractions. Tiie data was used to develop * a set of base curves relating the backwater ratio (h,/ & h) and the contraction ratio jvi for different values of roughness These base curves were obtained using vertical-faced con- strictions v/ith square edged abutments, ani a Froude number of 0.5- Auxiliary graphs were presented to modify the result according to the geometry of the constriction if other than the basic geometry. Vallentine ' used vertical sharp edged constriction plates placed normal to the flow. The discharge through the constriction was related to the upstream' depth by an equation containing an experimental discharge coefficient: Q} = dn b vZq *j ^ • ^ke variation of this discharge co- efficient with the Froude number of the unconstricted flow was given for several values of the contraction ratio. (7) jNagler did considerable experimental work with three dimensional models of bridge piers. Although he worked at an extremely small scale, several curves are presented re- lating the backwater to tne geometry of the pier. Husain carried out two ani three dimensional tests of archei openings on a small scale in preparation for a larger study of which the work reported here is a part. The prob- lem was also approached by dimensional analysis. General profiles of the backwater curves were obtained ani recom- mendations were made concerning the design of the future equipment . ( 2 ) oooky^ ' carried on the small scale testing program be- gun by Husain and included two and three dimensional models with and without channel roughness. For the two dimensional case, equations relating the discharge to the geometry of the opening ana the backwater height were developed. Curves were obtained from experimental data which relate the coef- ficient of discharge and the backwater height to the Froude number of the unconstricted flow and the contraction ratio. These equations are presented in Appendix B. DESIGN AND CONSTRUCTION OF THE APPARATUS Testing; Flume Preliminary Computations Before the design of the flume itself could proceed, it was necessary to determine whether the backwater and regain phenomena could be represented to a convenient and easily measurable scale in the space available in the hy- draulics laboratory. Several sets of arch bridge plans provided by the Indiana State Highway Department were analyzed for the values of backwater. The theory of varied flow and the equations and tables presented by Bakhmeteff ^ J and by the (a) U.S. Bureau of Public Roads w/ were used. It was only pos- sible to make an approximate calculation of the backwater because of the unknown effect D f an arch-type constriction. A sample calculation is presented for bridge S79 (Clay County , Ind i -ana ) . Assumption 1) Velocity through the bridge constriction = 6 feet/second. This value was suggested by Mr. J. I. Perry, Chief Engineer, Indiana State Flood Control and Water Resources Commission, as an average flood flow vel- ocity through typical bridges in Indiana. From bridge plan .Vaterway area through the bridge utilized by the 2 stream = 125 feet Slope of the stream bed = 0.0019 feet/foot Flow = 750 cfs The total waterway area under the arched constriction of the bridge was approximated by a trapezoid. (See figure 1 appendix G) The unconstricted area of the waterway up- 2 stream of the bridge was found to be 300 feet . In order to find the percentage of constriction of the stream it was necessary to proportion the flow within the subsections of the original area. The tabulation for the prooortionment is shown below. Sub n 1.486/n A P 3 Section feet 2 feet feet 1 .05 29.3 45 12 3.75 2 .05 29.8 210 30 7 3 .05 29.8 ^5 A = 300 12 3.75 Sub R 2/3 AR 2 ' 5 K Q Section cfs cfs 1 2.41 108.5 3,240 86 2 3.65 776.0 22,800 578 3 2.41 108.5 3,240 86 K = 28,280 Q = 750 The prooortionment was r nade on the basis of conveyance. The interface of sub sections was considered as a boundary. 7 p The total water-way area through the bridge was 143.5 feet located entirely in sub section 2. The loss in capacity based on relative conveyance was computed as follows. 2 Area deducted section 2 = 66.5 feet (see figure 1 aopendix C) 2 Area total section 2 = 210.5 feet Flow deducted section 2 = (66.5/210) x 578 = 182 cfs Flow deducted section 1 = 86 cfs Flow deducted section 3 = 86 cfs Total flow deducted = 35-4- cfs The loss in capacity represented by this constriction is 554- cfs. The contraction ratio M then is (354/750) x 100 or 47/y. The following computations are based on data taken from the U.S. Bureau of Public Roads report. ' Assuming the geometry of a 30° wing wall normal cross- ing, the backwater coefficient K, = 1.075 for a contraction ratio M = 47% V 2 /2g = 36/64.4 = .56 feet h-j^ = 1.075 (.56) = 0.602 feet. This is the maximum back- water superelevation. Length of Backwater Profile. The backwater profile was comouted using Bakhmetef f ' s^ ' function y \V; • The compu- tations made on this basis neglect the regain of kinetic energy in an expanding stream. In order to use the tables of 0\?7) , the hydraulic exponent n characteristic of the channel upstream of the bridge was computed by use of the equation n = 2 cot oC . (. <tC is shown in figure 2 ap- pendix C) y = 3 feet y 1=7 feet A = 90 + 18 = 108 feet 2 A x = 300 feet 2 p = 30 + 12 = 42 feet 2 p, = 44 feet 1 1 R - 108/42 =2.57 feet R ] _ = 300/44 = 6.82 feet «J = 1.486/. 05 ^2.57; 1/6 - 34.8 C 1 = 1.486/. 05 x (682) 1/6 = 41.0 K = 108 x 34.8 x 1.6 = 6000 iL ± = 300 x 41 x 2.61 = 52,100 Log K = 3-78 Log K x = 4.506 Log y = 0.478 Log y ± = 0.846 n = 2(4.506 - 3.78/0.436 - .478) = 2(. 726/. 368; = 3-95 y 7] *0 i L feet feet feet 7.602 1.086 0.660 7.500 1.070 0.599 .061 628 628 7. 400 1.058 0.544 .055 565 1193 7.300 1.041 0.446 .098 1010 2203 7.200 1.030 0.362 .084 865 . 3068 7.100 1.013 0.137 .255 2320 5388 7.050 1.00S 0.004 .133 1370 6758 7.020 1.003 0.264 .278 2860 9618 Original normal depth = 7.00 feet backwater = 0.60 feet Depth at maximum backwater = 7-60 feet Superelevation decrease = 0.58 feet This reach then includes 0.58/0.60 x 100 or 96% of the superelevation. Complete information is lacking on the length of the regain curve. It may be assumed that the angle of expansion of the stream downstream of the contraction may be approximated by the divergence angle of a submerged jet, and that the re- gain curve will be complete when the expansion has reached the full width of the flume. Albertson^ ' found a diver- gence angle on each side of the C3nterline of a free jet of 11 to 14 degrees. Henry ^ ' found the free boundary to di- verge at approximately 7 degrees for the flow from a sub- merged sluice gate. An angle of 5 is estimated with this particular bridge in order to obtain an approximation by excess of the regain curve as follows. (See figure 3, ap- pendix C). Clear soan of bridge at spring line = 30 feet. burface width of arch at maximum high water = 11 feet. Average width of opening = ( 30 + ll)/2 = 20.5 feet. Stream width = 46 feet The regain to normal depth should therefore be complete in a length of 146 feet measured from the downstream side of the constriction. This approximate computation indicates that the total length of the backwater curve reach (within 0.02 feet of the normal depth) and the estimated regain curve plus the bridge is 9618 + 146 + 48 = 9312 feet. It is now desired to find the required length of flume to rep- resent to scale the totality of the regain curve and a 10 reasonable portion of the backwater curve. The water flow available in the hydraulic laboratory was 2100 Gpm. 2100 Gpm = 2100/449 =4.7 cfs. 5/2 bince <^m/Qp = L the required scale ratio is found to be 4.7/750 = L 5/2 = 0.00627- Lr = C-00627J 2/5 = .155 or Lr = 1:7.4 The nearest convenient scale is 1:10. For this ratio, the superelevation = .06 feet = .72 inches. The usable length of the flume in the proposed soace would equal 60 feet. For this ratio, 600 feet of the prototype stream could be represented. This would include the totality of the regain curve, the bridge model and some 400 feet of the prototype backwater curve. From this and similar computations as well as small scale tests, ' it apoeared that a flume utilizing all of the easily available SDace in bhe laboratory that is a flume length of 64 feet but capable of extension would be satisfactory. The width of the flume w?s fixed at 5 feet. This was based on a consideration of the scale ratios and the space available. The cross section of the flume was to be rectangular since this configuration lent itself well to both ease of construction and adaption. Design Procedure In order to test the flows under varying slope con- ditions, the flume was to be tiltable about one end. 11 Several methods are available to achieve slope control including: 1. Hydraulic Jacks 2. Screw Jacks J. wedges 4. Sables and vV inches bcrew Jacks were selected because of accuracy and ease of control, as well as permanence and appearance. At the time the preliminary design wis made, only an estimate could be made of the final weight of the flume and the water contained therein. in order to keep the deflections due to bhe variable weight of water within the same order of magnitude of the smallest readings of the point gage for depth measurement, 0.1 mm, the flume bottom was designed of 1/4 inch steel plate supported at 2 foot intervals on channels. The channels in turn were to be supoorted by two or more main beams riding on the jacks. The sile plates were designed of 1/4 inch steel plate supported by vertical angles resting on the channel members. A longitudinal horizontal angle mounted on the vertical angles served as a support for the guide rails. The guide rails, which serve as a reference plane from which measurements are based, were to be nolished stainless steel to minimize corrosion and scale. The preliminary design was based on a possible water depth of 2. feet. In the immediately proposed tests this will allow a freeboard of approximately 1 foot. However, de- flection will be within the set limits for a loading of 12 2 feet of water which may 0C3ur at a later date. A portion of the first design is presented. Dead Load 1/4 inch plate 2 side plates 64 f set long x 2 faet wide @ 20.4 lb/foot = 2620 lb. 1 bottom plate 64 feet long x 5 feet wide © 51.0 lb/foot = 3260 lb. 5880 lb. Main Beam (18 I 54.7) 2 x 64 feet x 55 lb/feet = 7040 lbs. Channels 53 x 5 feet x 8.2 lb/feet = 1360 lbs. 14,280 lbs. Extras @ 30% 4,300 lbs. total 18,580 lbs. For design purposes this is an average load of 18,580 lbs/64 feet = 290 lbs/foot. Live Load. At the maximum depth of 1 feet the volume of water contained is 10 feet per foot of length. This imposes a load of 10 feet^ x 62.4 lbs/foot^ = 624 lbs/foot. The total load per foot then is 290 pounds/foot + 624 lbs/foot = 914 lbs/foot. The distance between supports was set as 20 feet. Since the exact nature of the main beam connections was unicnown, the solution was made based on a simply supported condition. Two alternatives were presented. The first was to use beams whose stiffness would make any deflections negligible 13 and the second was to use lighter beams and correct for t 3 deflections by adjusting screws. The first alternative was chosen as the second would necessitate adjustment after each change in conditions such as slope, or water iepth. The beam first selected was an 13 I 5^.7 which gave a calculated deflection of 0.00225 feet under the design loading. Contacts with the fabricator and erector were made at a later date and it was found that a 20 I 65.4- would be available at a cost less than that of the lighter beam. The use of the heavier beam w:s accepted and the design oroceeded based on this beam. The deflection due to the variable water weight was approximately 0.002 feet for a deoth of one foot. It was felt that some form of transverse leveling was necessary. Adjustment bolts were an obvious solution but the location was yet to be selected. The first sketches in- corporated adjustment bolts between the channel and the bot- tom plate. This produced an indeterminate situation with regard to flexure, inconvenient locations for adju-tment, and high fabrication cost. The subsequent designs placed the adjustment bolts between the channels and the main beams which gave the botcom plate full support across its width at 2 foot intervals. The bottom plate was designed slightly wider than the flume width. This permitted attaching an angle to hold the bottom edge of the vertical plate fixed. The upper edge of the vertical plate had nuts welded on at the two foot points. Studs were attached between the nuts and the 14 vertical angles to suo 'ort the plate and nrovide an ad- justment for its longitudinal alignment. The inside of the flume was finished with an epoxy resin apdied with a hand roller. The flume construction is shown in figures 4 and 5. ( Appendix C) In operations of this kind, guide rails ar^ generally fixed in the level position. They then may be used to con- trol the slope. oince the rails were attached to the flume, direct slope measurements was not possible. instead, dif- ferences between the surface of a standing body of water and the flume floor were used to measure the slope and calibrate a revolution indicator which served as the primary method of slope control. Tailgate Control over the depth was exercised by a gate mounted at the end of the flume. The gate was manually operated from the side of the flume. Figure 6 (apoendix C) shows the gate. The gate was made in such a way that it could be used either as a sluice or as a weir. Throughout this first series of tests, the gate was used exclusively as a weir. Forebay The forebay, 8 feet wide and 10 feet long, was con- structed of plywood and lined with sheet metal, and is shown in figure 10 of apoendix C. The 3 inch and 6 inch pipes entered the rear of the forebay at the top. The 6 inch 15 line was centered and the 3 inch liiEwas placed slightly off center. The diffusing mechanism for each supply line con- sisted of a tee and cross pipe of the sane size as the line at the bottom of the forebay. The turbulence of the entering water was controlled by a 4 inch gravel baffle and three wire mesh screens of 13 mesh per inch. The transition section con- tinuing into the bottom and side walls of the flume was made of quarter ellipses in the horizontal and vertical planes respectively with a ratio of major to minor axes of 1.5 to 1.0. The joint between the flume and the forebay was sealed with a flexible rubber gasket mounted so as not to interfere with the flow. Slooe Control Mechanism The flume rests upon six screw jacks and a hinge. The hinge is located at the joint of the flume and the forebay. The jacks are similar in all respects with the exception of the gear ratio. The jacks are divided into three pairs with rates of raise of one, two, and three inches for 96 turns of the shaft. aince the hinge was a fixad point and the opposite end of the flums was the point of maximum movement, the jacks were arranged such that the pair nearest the hinge moved the least and the pair at the opposite end of the fluma had lar- gest displac 3mant per revolution. This maintained the bottom of the flume as a plan? while it was baing raised and lowered. The jacks on each side of the flume were driven by a common 1 inch shaft line connected at one and to a 90 miter gear. The miter gears on ei^har side in turn were connected 16 to a single 60:1 ratio gear reducer. The power to operate all the jacks was supplied by a 1-1/2 horsepower, 1750 re- volutions per minute, reversible, electric motor connected directly to the gear reducer. This provides a rate of ver- tical displacement at the dov/nstream .lack station of ap- proximately 1 inch per minute. The jacks were arranged in such a way that the downstream end of the flume may move from 12 inches below hori .ontal to 3 inches above horizontal, resulting in a maximum positive slope of 1/60 and a maximum adverse slope of 1/240. The motor was controlled by a raise, lower, and stop control switch. Safety switches were located both near the motor and near the control switch. It was necessary to unlock these before tne control circuit could be completed. In addition, automatic limit switches were provided to prevent running the jacks beyond their limits. The general arrangement of jacks and gears is shown in figure 7» In order to connect the ends of the jacks (which move in a vertical line) to the flume (which moves in an arc), it was necessary to use a pinned linkage. A photograph of the linkage is shown in figure 8. (Appendix G) Due to the arc of the linkage, the relation between the rise of the flume and the revolutions turned by the jack shafts was not linear. Therefore, it was necessary to make • a calibration of bhe slope rather than computing it. At the time of erection, the jacks were leveled at .001 foot before the flume was erected. During the alignment oro- cedure the ja^ks were raised or lowered individually as needed 17 to obtain a level base. The shaft couplings were then in- stalled an no further individual movements were made. Construction Considerable time was spent in obtaining the requi- sitions, bids from several contractors and actual super- vision of the erection of the flume. The foundations, jack piers, plumbing, and electrical controls were installed by Purdue university Physical Plant. The structural parts of the flume were built and assembled by a contractor, the flume adjustments, construction of the instrument carriage, installation of manometers and calibrations were done by the Research Assistant and student labor »vhen needed, 18 PREPARATION FOR TESTING Slope Calibration After the flume was aligned and leveled a slope cal- ibration was made by visually counting the revolutions of the slowest speed shaft and measuring the depth of a still ■dooI of water at two points 50 feet apart. A steel tape was installed with station at the upstream end and station 64 at the downstream end of the flume. The points chosen for slope measurement were stations 6 and 56. These points had previously given consistently good results when measurements were made of the distance betwe n the rails ani the flume floor. The calibration of slope versus revolutions appears in figure 9« (Appendix 0) The apparent scatter of the points toward the downstream end of the flume is due to the magnifi- cation resulting from the logarithmic scale at that end. In order to avoid the necessity of visually counting shaft revolutions to keep track of the sllope, a revolution indicator was made and installed at jack station number 3« The lowering of the flume activated the pointer which both multiplied and reversed the motion. The tip of the t>ointer rode on a lucite strip mounted below the motor controls. A mark was scribed on the lucite strip at each revolution over the range through 40 as well as at the test slopes. On the end of the shaft a circular lucite plate divided into ten 19 parts was mounted along with a fixed, pointer. Dhe slope desired was set by using the large indicator to the nearest revolution and setting the tenths of a rsvolution by using the small dial. This equipment was liter replaced by a com- mercial revolution counter mounted at the same location. This counter read directly to a tenth of a revolution and the reading could be interpolated to one half of that. This device provides a slope control with an accuracy of + 0.0000025 feet/feet. Figure 8 (Appendix G shows the counter) vVater Supply and Measurement System The water available in the laboratory is recirculated through the system by two pumps rated at 300 Gpm and 2000 Gpm. The 3 inch discharge line from the 300 Gpm pump was connected to a new 3 inch line. This line contained a new 3 x 2,25 inch venturi accurate to 0.5% over the range from 30 Gpm to 300 Gpm. The line was installed using long sweep elbows to reduce head loss. A 60 inch differential manometer reading to 0.01 inch was connected to the venturi and filled with tetrabromoethane (Specific gravity 2.95) which gave a man- ometer deflection of 51 inches with a flow of 336 Gpm. The 2000 Gpm pumo was connected to an existing 6 inch line which was improved by the installation of long sweep elbows in place of tee's and short elbows. In adlition, a new 6 x 4.176 inch venturi accurate to 0.5% over the range 200 Gpm to 2000 Gpm was installed preceeded by a set of straightening vanes. A 30 inch differential manometer reading 20 to 0.01 inch was connected to the venturi and filled with mercury (Specific gravity 13.6). This manometer gave a de- flection of 14.9 inches for a maximum flow of 1790 Gpm. In both cases the Venturis were fitted with air vents to insure proper measurements. After a portion of the tasts had been made and the data evaluated, it was deemed necessary to improve the discharge measurements. The 60 inch manometer was attached to the 6 inch venturi and recalibrated using tet- rabromoethane as the manometer fluid. This resulted in a larger manometric deflection improving the accuracy of the measurement. The 50 inch manometer was connected to the 3 inch venturi but thare was no need to recalibrate the meter. In order that the calibration of the venturi meters should have no error larger than that of the venturi meter, the scale to be used for the calibration was checked against standard weights by the indiana State Board of Health, Division of .'."eights and Measures. The scale error was less tnan 0.2-b or 2 pounds per 1000 pounds. For the purpose of calibrating of the venturi meters, branch lines led to a baffled concrete channel located above the weighing tank. At the point immediately before the 3 inch ani 6 inch lines entered the forebay, valves and valve bypasses were installed. The 6 inch line had a 2 inch bv-pass and valve and the 3 inch line was fitted with a 1 inch bypass and valve. The manometers were mounted in a position easily visible to the person adjusting the valves. The overall apparatus arrangement in the laboratory is shown in figure 10. (Appendix G) 21 Venturi Calibration As soon as the essential piping was completed, cal- ibration of the venturi meters was begun. The procedure was as follows. The flow was set using a valve located downstream of the venturi and a waiting period of aporoximately 5 minutes was allowed foe the system to come to equilibrium under the new flow. ihe scales were preset to an arbitrary weight and the weighing tank valve closed. The manometer deflection was noted ani the water diverted into the weighing tank. The scales wer? tripoed and the timer manually started 7/hen the weight of water collected equalled the weight which had been preset on tne scale beam. Tne scale weight was noted. This method of calibration avoids the errors of non instantaneous starting and stopping of flow but still does not correct or make allowances for the difference in tne impact force of the water entering an empty tank as compared to a partially full tank. This er?or is of the magnitude of 1% which is com- patable with accuracy of the remainder of the system. fhe intervals oi' calibration were selected so as to fall between one and two inches of deflection on the 60 inch manometer containing the lighter fluid and not to exceed 1 inch on the 30 inch manometer containing the mercury. The calibration curves are presented in figures 11, 12 and 13- (Appendix J) 22 MODEL TESTING model Construction From results of the preliminary experiments, (figure 14- Appendix C) it was found that the predominant variable was the contraction ratio and that the length of the model had little or no influence for Froude numbers less than 0.5. This range of Froude numbers corresponds to the case usually found in practice. It was therefore decided that the first series of tests would concentrate on two dimensional sharp edge semi-circular models with no skew. The cost of machining mild steel plates to produce the constrictions was prohibitive. An estimate was obtained of 8125.00 per model. The frequency of handling and changing the constriction plates also necessitated a model that wis light weight and still capable of sustaining hard use without damage. The models, as finally made, consisted of back up sheets of 1/2 inch exterior grade plywood faced v/ith a sheet of 22 gauge galvanized iron and braced by a steel angle. The openings were cut out of the galvanized iron sheet accurate to 1/32 inch. The ooening in the plywood backing had a radius of 1/2 inch greater than that used in the metal. The metal was bolted tightly against the wood. In the flume, the model was positioned with the metal face upstream. The construction of the models is shown in figure 15. (Appendix C) 23 Four models were made with contraction, ratios of 0.3, 0.5, 0.7, and 0.9. At the slopes and flows tested, the model with m = 0.3 was submerged on a majority of the tests. The data oresented in Appendix G is that collected for the other three models which are shown in figure 5- (Aopendix C) Free Surface Measurement The position of the free surface was measured with an electric indicating point gaga reading to the nearest 0.1 mm. The point gage was mounted on an aluminum and brass bar in such a way that the gage could traverse the width of the flume This bar in turn was part of a carriage which rolled on the stainless steel rails along the flume. The carriage was rectangular and rolled on 4 wheels, two of which on one side of the carriage were grooved to provide alignment. The power supoly was mounted at the back of the carriage, The operator rode on a second carriage which was on its own rails. Details concerning the instrument carriage can be seen in figure 16. (Appendix C) The head of the point gage was com prised of two probes approximately 1/4 inc:h in diameter. The probes were sep- arated a distance of about 1-1/2 inches. The rear electrode was 15 millimeters longer than the front, and served as the ground. The end of the front electrode was pointed and was adjusted to the water surface. in operation, the obstruction to the flow presented by the rear electrode caused a rise in the water surface of v /2g against the electrode. The effect of this disturbance extended upstream to the measuring probe 24 and made it impossible to measure the true water surface. This effect was not a constant since the velocity head was different in each case. In order to improve the performance of the instrument, the rear electrode was replaced by a small diameter copper wire. This wire presented a much smaller obstruction to the flow and the operation of the gage was not only simplified, but gave data that should be superior to that obtained with the former arrangement. Uniform Flow Calibration Preliminary tests were run to determine uniform flow conditions in the flume. The variables involved are: the discharge, the slope and the tailgate settings. For each flow and slope the tailgate setting was determined by trial and error until uniform depth was obtained along the largest pos- sible reach of the flume. The models were located in such a way that the regain curve was complete within the uniform depth section, effects of boundary layer growth were not considered, and the flow was considered uniform as long as the depth remained constant. As the model tests progressed the uniform flow con- ditions were recorded in the auxiliary graphs of figures 20 and 21. (Appendix J) The first was the normal depth versus the slope and the second was the height of the tailgate versus the slope. In both cases, the rate of flow was the parameter. In order to obtain data that would provide a complete coverage of the 25 range of Froude numbers investigated, the Froude number was first chosen and the normal depth computed in the following manner. f c v a or tt 3 . Q* /T 2 3 Vf Gf x '/3 from which " [Fo* B VJ After the desired normal iepth was computed, the slope for a particular flow which would give this value could be determined from figure 20 (Appendix 0). Then, entering fig- ure 21 (Appendix G) with the slope and flow, the necessary gate setting to produce uniform flow could be determined. Testing After the slope and uniform flow calibrations were com- pleted the testing was begun. The procedure used was to set the flow, which was the most difficult quantity to adjust, and let it remain constant while the slope, tail gate setting and models were changed. For each case of slope an I flow, the tail gate heighth was set according to the previous un- iform flow calibrations. In the majority of cases the con- ditions of uniform flow was obtained on the first trial and in every case no more than two trial gate settings were needed. Once the system had come to equilibrium and the 26 normal depth was obtained, a model was installed in the flume. For the first few runs, the complete profile was taken. It was observed that in the latter oortion of the regain curve, the water surface fluctuated vertically as much as one centimeter from one minute to the next. This fluctuation did not occur rabidly but rather slowly. Testing was immediately suspended while the cause for this was deter- mined. The first oossible cause investigated was tnat of a variable inflow into the flume. However, observation of the backwater showed it to be very stable. This indicated that the flow into the flume was constant and the phenomena was due to the model or that portion of the flume beyond the model. Eddies were suspected as a possible cause of this phenomenon. In an effort to eliminate eddies, the gap between the metal facing and the plywood back up board was filled and beveled to oroduce an angle of 4-5 . This resulted in no appreciable change. The condition most suspect however, was channelization of the flow to one side of the flume with eddies on the opposite side. It seemed that an instability was developed by the constriction. The reason for this channelization of the flow is not known. Misalignment of the bottom was not a factor because it v/as level within 1 millimeter throughout the length of the flume. The tail gate was leveled to an accuracy of 1 millimeter an i the models were installed, using a square to check the alignment both horizontally and vertically. It was discovered that in a given case with most of the flow on 27 one side, after artificially deflecting the flow to the other side, it would remain on that side. This suggested that the lack of symmetry was not a factor. One possible solution is the addition of roughness and increasing of slope to sta- bilize the flow. However, since these tests were to be specifically smooth boundary tests, this was not done and the affected portion of the regain section was neglected. For the remainder of the tests, only a short profile before and after the model was taken to locate the points of maximum and minimum depth. Figures 17, 18, and 19 (Appendix 3) are photographs of the flume with model in place. Figure 18 shows the flow going to one side and figure 19 shows the flow centered. Test Results The test data and the calculated values of the dis- charge coefficient CL., of the friction coefficient f, and of the Reynolds number are presented in table I. The ratio of the backwater depth to th3 normal depth y, /y is plotted versus the ratio of tne velocity head to normal depth with the contraction ratio m as a parameter in figure 22. The ratio of the velocity head to the normal depth is a measure of the kineticity of the flow, it is exactly half of the kin- eticity as defined by Bakhmetef f v yj , or half the square of the Froude number. The discharge coefficient C~ calculated from equation 2 of Appendix B is plotted versus the ratio of tne velocity heal to the normal depth with tne contraction ratio m as a parameter in figure 23. 28 The consistency of the data is well illustrated by the lack of scatter of the experimental points as plotted in fig- ures 22 and 23. (Appendix C) These test results may be com- pared with the small scale tests for the contraction ratio of 0.5 which is common to both test series. Inspection of figure 14- (Appendix 0) and figures 22 and 23 (Appendix G) show that the values are almost identical. For example, at a Froude number of 0.2, the value of the discharge coef- ficient Cj. from the small scale tests (figure 14- Appendix C) is 0.38 and the superelevation ratio y, /y is 1.1. Compared to this, the large scale tests, (Figure 22 and 23 Appendix C) give G~ as 0.275 and 7-,/y as 1.119 for a kin- eticity of 0.02 which corresponds to the Froude number of 0.2. At a Froude number of 0.4- the results of the small scale tests indicated G„ was equal to 0.53 and y, /y was 1.4. Similarly, at a corresponding kineticy of 0.08, the large scale tests showed G n to be 0.4-83 and y, /y to be 1.4-32. The reliability of the data may be better discussed in terms of the values of the friction coefficient and of the backwater ratio. The Darcy-Weisbach friction factor and the Reynolds number for the uniform flow established before each test were calculated in table I as follows: v r = 89 RS 29 The experimental friction factors were compared to the theoretical values obtained by adapting the Blasius and Prandtl formulas for flow in smooth pipes to the rectangular open channel. Th9 formulas for smooth pipe flow are: Blasius f = 0.3/6^ (W) -% R</0 S Prandtl-Von Karman _J_ - £ Q /oo fVJ? \Tf) -O.8 Replacing D by 4R and simplifying, the equations become: Blasius f~- O.Z23 (^) V * Prandtl-Von Karman -J^ =Z.O /oa ( VR ]ff ) +OAO (12) '^ V Powell and Posey v J , working with a triangular flume found the formula governing their friction factor to be / = 2.074- /o 3 (/RfT) -0-797 for tranquil flow in a smooth channel. The comparison between experimental values and the theoretical formulas is shown in figures 24 and 25 • (Appendix G) In figure 24 the ex- perimental values of the Darcy-Weisbach coefficient f are plotted versus the Reynolds number, both f and fR are as defined above based on the hydraulic radius. Also plotted on the same figure are the Blasius and Von-Karman relationship as well as the general range of experimental values obtained by Lansford and Robert son v ' J for smooth triangular channels. In figure 25 (Appendix C) the friction coefficient f and the Reynolds number are basad on the normal dapth, assuming two dimensional flow. Although this assumption is not completely true for the depth to width ratio used in 50 the experiments this was done to comoare the data with the (140 experiments of Owen v ' which were done in a glass channel. In general, as shown in figure 24, (Aooeniix 0) the data fall above the theoretical lines which are a lower limit for a perfectly smooth boundary. The average friction co- efficient is f =0.021 which corresponds to an absolute roughness of approximately £ = f (f) = 0.0025 feet. This corresponds to the irregularities of the epoxy resin of the channel finish. The percentage of probable er ^or of the friction co- efficient is calculated as follows. The calculations are made based on a flow of 5 cfs ani a slope of 0.000100 foot/feet, The measurement of the flow during calibration had a possible er :or of 1% due to neglecting the impact in the weighing tank with different water depths, and the scales had a pos- sible error of 0.2%. The possible error in the Venturis was 0.5% and the observed manometer readings could have been in error by 0.02 inches. With the given flow, the manometer deflection was 52 inches of tetrabromoethane . This means that the possible error in reading the deflection value was approximately 0.02/54- or 5.85%. Since the flow varies with the 0.5 power of the deflection, this would represent an error in the flow of 1.95%. Therefore, the total possible er^or in the flow is 5.65%. The slope was calibrated and set to less than one tenth revolution of the connecting shafts. A maximum error of one twentieth of a revolution is equal to 0.0000025 feet/feet. At the slope 31 chosen, this would produce an error of 0.0000025/. 0001 or 2.5%« The measurement of y was made to 0.1 millimeter. An error of this magnitude with the value of y found for this condition (21.70 cm) amounts to 0.01 cm/21.70 cm = 0.046%. The error in computing the wetted perimeter could be 2(0.01 + 0.5)cm/192 cm = 0.053%. Since f = 8gRS/\/ 2 or 3 2 8gSy /Q p, the error in f can be expressed as V 2.5 2 + (3 x 0.046) 2 + (2 x 3.63) 2 + (0.053) 2 = '58.97 = 7.68. At flatter slooes or lower flows, this error would be even larger than calculated. Oonversly, those tests at steeper slopes and higher flows should give the most nearly correct values of f. however, the points T^ould not be ex- pected to fall on the theoretical line since the materials used in the construction of the flume and the finish applied certainly caused some finite value of roughness. A. second check was made by determining the backwater ratio h,/4 h for each model test and comoaring the values ( 6") obtained to those presented by Tracy and Jarter v } for rectangular constrictions. This comparison is shown in figure 26 (Appendix C) which shows that the backwater ratios are of the same order of magnitude although different as may be expected with different boundary geometries. 32 SUMMARY OF RESULTS /LNU CONCLUSIONS Flume Design The flume, as designed and built has proved adequate to carry out the proposed experiments. The entire apparatus necessary to carry out th? testing program can be run con- veniently by two men. The slope controls, including the jacks ani motor, function well and changing slope takes less than 5 minutes. Testing Only the first series of tests, sharp edged, semi- circular constrictions, have been completed on a large scale so comparisons cannot be drawn as such. However, several things indicate the validity of the lata. 1. Close agreement with the small scale tests. 2. Comparison with other investigators on: a. Friction factor. b. Mannings roughness 3. Adherance of the plotted lata to a well defined pattern with little experimental scatter. The conclusions possible so far are primarilly drawn from figure 23 which relates the lischarge coefficient and the kineticity of the flow, and figure22 which shows the backwater in terms of kineticity. From the first, it can 33 be seen that above a value that corresponds with a Fronde number of 0.5, the Cj, value ceases to depend on the kin- eticity of the flow. The second, figure 22 shows the de- pendance of trie backwater value on both the contraction ratio and the kineticity. 34 BIBLIOGRAPHY 1. ilusain, 3. T., "ir'reliminary WiOdel Investigations of Hydraulic Characteristics of River Flow Under Arch Bridges" Masters Thesis, Purdue University, 1959. 2. Owen, H. J.; Sooky, A; Husain, S. T.; Delleur, J. W. ; " Hydraulics of River Flow Under Arch Bridges - A Progress Report." Progress Report submitted to the Board of the Joint Highway Research Project, May 14, 1959. 5. Lane, E. W. , "experiments on the Flow of Water Through Contractions in an Open Channel ' Transactions ASCE Vol. 83, 1919-1920. 4. Kindsvater 0. E.; Garter, R. W. ; "Tranquil Flow Through Open Channel Constrictions". Transactions ASCe, Vol. 120, 1955- 5. Tracy, H. J.; Carter, R. ft'.; "Backwater Effects of Open Channel Jonstrictions" . Transactions, ASCE, vol. 120, 1955. 6. vallentine, n. R. , "Flow in Rectangular Channels with Lateral Constriction Plates". La Houille Blanche, January, 1958. 7. Magler, F. A.; "Obstructions of Bridge Piers to the Flow of water". Transactions ASCE, Vol. 82, 1918 PP354-95- 8. Bakhmeteff, B. A.; "Hydraulics of Open Channels". Engineering Societies monograph, McGraw-Hill Series, 1932. 9. U.S. Bureau of Public Roads, "Computation of Back- water Caused by Bridges". October, 1958. 10. Albertson, m. L. ; Dai, Y. B.; Jensen, R. A.; Souse, H.; "Diffusion of Submerged Jets" Transactions, ASCE, Vol. 115, 1950. 11. Henry, H. R.; "Discussion of 10 " Transactions, ASCE, Vol. 115, 1950. 35 12. Powell, R. W.; .Posey, C. J.; "Resistance Experimants in a Triangular Channel". Journal of the Hydraulics Division, Proceedings ASGE, May 1959. 13. Lansford >v. ivi. ; Robertson J, M. Discussion, trans- actions Ao"E, Vol. 123, 1958 p. 707. 14. Owen, W. to. ; "Laminar to Turbulent Flow in .Vide Open Channel" Transactions ASCE, Vol. 119, 1954. APPJSADIX A NOTATIONS 36 NUTATIONS SYMBOL UimIT DEFINITION 2 A feet Area of flow. B feet Stream width at bridge site or flume width. b feet Width of constriction opening equal to di- ameter of semi-circle. G-pj Coefficient of discaarge equation. D feet Pipe diameter. E An infinite series of powers of y /r. t o F Froude number of flow. Subscript o refers to unconstricted stream. f Darcy-Weisbach friction factor. g feet/sec Acceleration of gravity. h, feet Superelevation of back- water above normal depth. h^ feet Boundary friction loss. A k feet or Elevation difference be- centimeters tween points of maximum 37 Lr M cf s feet feet feet feet cfs cfs and minimum depth. Backwater head loss co- efficient . Conveyance of a channel or section of channel = AG \fR. Length of reach in back- water computations. Accumulated length in backwater computation. Scale of length. Channel contraction ratio equal to 1-b/B (in review of lit- erature) . Channel contraction ratio equal to b/B. Manning roughness or hydraulic exponent in K = const, y Wetted perimeter of stream or subsection. Total flow (subscripts m and p refer to model and prototype). flow in a subsection (subscripts m and p 38 refer to small model flume and to prototype flume. ) R feet Hydraulic radius. r feet Radius of semi-circular constriction. nn Reynolds number. S feet/feet Slope of stream bed or flume. V feet/sec Average velocity (sub- script o refers to un- constricted flow). y feet or Depth of flow. centimeters y feet or Normal deoth of flow in o centimeters unconstricted channel, y-, feet or Maximum depth of flow centimeters upstream of constriction. c<_ Slope of line on Con- veyance versus Depth graph. P Ratio of bottom slope to critical slope. 7f ' y/y o Bakhmeteff backwater function. APPENDIX B EQUATIONS Oi' FLOW 39 UJiRIVATi-OlMb OJB' i^UATIOrJS GOVERNING THE PLOW IN RECTANGULAR CHANNELS WlTn fc>EMl-0±RCULAR CONSTRICTIONS The equation for the discharge in rectangular channels with a sharp crested serai-circular constriction is ob- tained and is expressed in terms of an infinite series of powers of the ratio y-,/r. With reference to Figure 27, (Appendix C) Bernoulli theorem gives V= CV23V - C V2 9M-h) The element of area is d A-- 2 Vr^TT 2 dh and the discharge is thus Q '/VdA *{*'CVZjttrh)-2h*-h* dh (i) Expanding into a series, integrating term by term, and making "use of the fact that 2r = b: Q-C D vgy g y* b[)-0.IW($f-QM&)$~3 '(2) This may be written as Q -- C ^ fc £ (?) where and E - {[-O.IZ?4(^f~ O.OI77(^)* -t ■■] (5) The discharge in the rectangular flume is given by q = VoAo = Fo *f & y* /z (6) where (7) 40 is the i'Toude number of the undisturbed flow. Equating (.2; and (6) and solving for the coefficient of discharge Co -^-A (%■)% where m = 4 (9 ) or 5o L /7 /77^"Cp J APPENDIX C FIGURES AMD TABLES 41 -SECTION 1 SECTION 2 SECTION 3- ) t W//\ KLJ / / / V/f ^< W//ffi 3' S^ { js ^lV f- h- 30- I UPSTREAM OF BRIDGE 30'- THR0U6H BRIDGE IDEALIZED STREAM CROSS SECTION FIGURE I 42 o o O O _] CONVEYANCE vs. DEPTH FOR BRIDGE S79 (CLAY COUNTY, INDIANA) 1 \ \ \ \ \ \ 1 \ L in \ 0) \ \ > \ \ \ \l \ \ \ \ o o ro — >i 901 ■o CM UJ q: CD 43 TO UJ tr (3 44 45 FIGURE 5 FLUME AND MODELS FIGURE 6 TAIL GATE CONSTRUCTION 46 3SIVH .1 SO oj «ao» jo SNuru 96 ry/ #13S »ovr ^ v Tt j S ft f-f©1- asi»a „2 aoj »bo» , Lr-^]^ 2#13S X r2 £ 3SIVS ,.C HOJ WHOM JO SNUHJ. 96 £ * 13S MDVr CO o < Ll O L±J 47 FIGURE 8 JACK DETAIL 48 ^ ( S> o ~o o 2 1 o u. SLOPE CALIBRATIO CURVE FOR HYDRAULIC TESTIh FLUME ( \ 1 .000010 oc SLOPE- \ < i < !\ t > O O ffi id S HI " f o — SNonnnoA3a 49 CD < cr CT < CO Z) < cr a. < V U2 50 .a " > u .5 / / .5 j f / 5 J / / 5 / / / / CAL BRAT REE- ION C INCH URVE VENT FOR URI TH (.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0*5 0.70 0.75 0.80 FLOW RATE; CU. FT. PER SEC. FIGURE II 51 16.0 15.0 14.0 13.0 CO rd - 12.0 ii ad o Q; M.O CO ccio.O O CE ^ 9.0 8.0 LU Q <r w i- UJ 7.0 6.0 5.0 O 4.0 3.0 2.0 1.0 CAL IBRATION CURVE FOR s 1 IX- IN CH VI :NTUF M 1 0.5 i.O 1.5 2.0 2.5 3.0 3.5 4.0 4.5 FLOW RATE; CU. FT. PER SEC. FIGURE 12 52 o < (J, l-lfi UJ wc> _) Oco "- 2 „ UJ O q cor mo g] iro.- fr t-tn uj w 5 »- o z < 2 2 - 7r i CALIBRATION CURVE 6" VENTURI / I 1 40 f° r / / ' / 10 / \ ^ 1 FLOW RATE; CU. FT- PER SEC. FIGURE 13 53 54 cr Z) CD 55 FIGURE 16 INSTRUMENT CARRIAGE FIGURE 17 MODEL IN PLACE 56 MODEL IN PLACE FIGURE 18 FIGURE 19 MODEL IN PLACE 57 \Z T ZZj // Z /' f 7 / / / 2L-- 5 i £ J -J U. i- 7 i- / : / 1 / (. / / f u I 3 b 1 < c 1 1 en 5 o o CY UJ 0- O _l UJ i- W3 1H9I3H 31V9 / / / / I u, / / / / / i 7 / / / / V / / 1 / \ / 1 ' / / DEPTH VS. SLOPE FOR N6 FLUME J 1 1 > 1 in t/i o o <y ro if) o _i < 2 c o z <n UJ i- 1J °A Hid 3d nvwyoN 58 1 6 I SUPERELEVATION V s KINETICITY 1 7 t> O- m = 5 ^ 1 6 s-m ■ -9 / y \ / 1 5 1/ / o / 1 / 'o 14 1.3 1 2 8 1 / / <i i/ L_^m=- 9 1.0 - S ^iri ■n n 0.02 004 O06\/ 2 , 008 0.I0 12 14 Vo I 2 9 ' y FIGURE ^22 59 c .50 .4 5 4 35 .30 25 20 15 .10 .05 h^ ° / / * / / / I / / / / / 1 / / / / / / 1 / / / / / / j f / i / i / 7 / / / I / DISGHARGF COEFFICIENT KINETICITY o-m ' 5 &-m - 7 Q-/77 -.9 ± / 77 / 7 / M 4 I u c 02 04 06 .08 .10 .12 .14 16 18 v. 1 , 23 y. 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